Ellinger Y., Defranceschi M. (eds.) Strategies and applications in quantum chemistry (Kluwer, 200
.pdf326 G. DE BROUCKERE
The correlation effects on the electric dipole moment (Table 5) are much larger at the MR
SD-CI level for the state while the HF SD-CI algebraic values differ slightly for the two states, the qualities of the respective reference spaces being comparable[8].We find the
following magnitudes' sequence, among the dipole moments and presently it cannot be verified by experiment. MR SD-CI correlation effects noticed on
the electric quadrupole moment are also larger for the |
state. These effects on the |
||||||
nuclear nitrogen couplings' z component for the |
state appear to be negligible with |
||||||
respect to their HF SD-CI homologues which is generally |
quite unusual |
in case of |
the |
||||
ground state. The absolute value of the |
coupling of the |
state, |
2.554MHz, |
is |
|||
smaller than that for the |
state, 2.635MHz, whereas the |
state nuclear coupling |
|||||
appears to be the smallest: this sequence seems plausible as the electric field gradient at the nitrogen nucleus should diminish as one goes to higher energy molecular excited states.
Energy (a.u.);
Entries |
and analoguously for the other components. |
|
|
'Generated' |
Total number of spin and symmetry adapted single and double excitations |
'Selected' |
Number of spin and symmetry adapted configurations selected by second- |
|
order perturbation theory and treatedvariationally |
'*' |
Property calculated with respect to the center of mass |
'Est.full CI En' Estimated full-CI energy [11]
SPECTROSCOPIC OBSERVABLES FOR THE PN AND STATES 327
Several spectroscopic constants derived with and without the Davidson correction (Table
6) show little differences except for and the overall agreement with experiment being satisfactory. As for the X state, the Davidson correction tends to reflect
experiment sometimes better, e.g. for |
whereas the discrepancy with respect to |
experiment is sizeably reduced for |
|
Entries
'Emin' |
Energyof the minimum of the potential curve |
'E0 ' |
Energy corresponding to vibrational quantum number v=0 |
'E(ZP)' |
|
'EXP' |
Experimental values for the |
band [22] |
'other' |
see [15] |
|
328 |
|
|
|
|
G. DE BROUCKERE |
|
A deviation of merely 0.5% is found for the experimental |
|
value. Most of these |
||||
results for both states compared favourably |
with those obtained by Grein and Kapur [15] |
|||||
using smaller basis sets and another CI algorithm |
also based on single and double |
|||||
excitations out of a multi-reference space. |
For |
and |
of the |
state our results |
||
(with the Davidson correction) are closer to the experimental values. |
|
|
||||
We find the |
state to lie at merely |
above |
the |
state. The |
electronic |
|
spectra exhibit strong similarities with PN and, in particular, the same sequence is
observed, the corresponding states being separated by only [22] which should make our PN value very likely to be observed.
3.2.2 Spontaneous radiative lifetimes for the |
|
bands and |
|
corresponding lifetimes for the |
and |
states |
|
Spontaneous radiative lifetimes via electric dipole transitions for the |
and |
||
transitions are summarized in Table7 for several sets of (v',v") values.
Experimentally, the detection of the so-called Hanle signal which consists of measuring the change of fluorescence intensity in a transition band as the magnetic sublevels separate in a variable external magnetic field provides a direct measure of the upper state lifetime [23] which has been performed only for v'=0. This value agrees reasonably with its theoretical counterpart. A lack of a still better agreement with experiment might be related to the
presence of the very nearby state perturbing selectively the level, no experimental analysis by high resolution spectroscopy [4] or through the Hanle signal
measurement [23] being known for the |
state (see also section 1.2). Whereas for the |
|
first transition band no rotational dependence for |
is noticed and the lifetimes |
|
increase with the vibrational quantum number, a small but clear-cut rotational effect does exist for v'=0 - 2 in case of the second system. Moreover these lifetimes decrease with v' (except for v'=4) in contrast with the previous system. These effects are related to the evolution of the respective transition moments as function of the internuclear distance [8].
The large difference in magnitude among the lifetimes of both systems are linked to the corresponding values of the Einstein coefficients [8]. These results have been compared with those issued from a theoretical study of the same transition in CO [24], isoelectronic in the valence shell's electrons: it appears that our PN lifetimes are of the same order of magnitude as those in CO. Because a monotonic decrease in the CO lifetimes from low v' up to high-lying vibrational levels has been predicted too which were found consistent with experiment for the latter levels, we believe our PN lifetime values should be of the correct order of magnitude.
A lifetimes' comparison for each excited state (Table 8) shows that they are smaller for
the |
state except for v'=0. |
The vibrational decays occur by means of cascade |
|||
processes. Rotational effects |
|
appear to be even more intense than for |
|||
the corresponding transition bands and in either case these effects disappear for |
For |
||||
both states the lifetimes decrease with upward v' values. For the |
state a comparison of |
||||
our results with those on CO shows that the |
lifetimes are larger by an amount |
||||
similar to that noted for the |
transition. Unfortuntely no conclusive lifetime |
||||
measurements for the |
state seem to exist nowadays such that no useful informations |
SPECTROSCOPIC OBSERVABLES FOR THE PN AND STATES 329
can be invoked from the CO molecule to subtantiate our corresponding lifetimes for the PN
state. |
|
|
Comparing the vibrational |
lifetimes issued from both decay mechanisms (Tables 7 - 8), |
|
it is readily seen that the electric dipolar |
transition decay is always slightly favoured. A |
|
similar conclusion holds for the |
state but, as expected, the vibrational transition |
|
probabilities are much larger |
for the dipolar decay which lead to much smaller vibrational |
|
lifetimes with respect to those via the cascade mode of decay, the differences amounting to five to six powers of ten (Table 7 - 8).
330 |
G. DE BROUCKERE |
3.2.3. Miscellaneous spectroscopic observables
The emission spectrum observed by high resolution spectroscopy for the
vibrational bands [4] has been very well reproduced theoretically |
for several low-lying |
vibrational quantum numbers and the spectrum for the |
vibrational bands has |
been theoretically derived for low vibrational quantum numbers to be subjected to further experimental analysis [8]. Related Franck-Condon factors for the latter and former transition bands [8] have also been derived and compared favourably with semi-empirical calculations [25] performed for the former transition bands. Pure rotational, vibrational and rovibrational transitions appear to be the largest for the X ground state followed by those
for the |
and |
states respectively [8]. Whereas accurate data on pure rotational |
excitations were available for the X ground state [2] with which a fairly good theoretical agreement was obtained (Table 3), no such data exist for these excited states. Because the three states have been treated identically in solving the corresponding one-dimensional
Schrödinger equations, the qualities of the respective reference spaces being rather close [7,8] and noting the algorithm provides exact solutions within the numerical procedure utilized, we believe the theoretical data should also be verified by accurate microwave
measurements. The experimental trend shown by the expectation values of the electric
SPECTROSCOPIC OBSERVABLES FOR THE PN |
AND STATES |
331 |
dipole moment over a few vibrational functions is well reflected theoretically for the X state [7]. Different patterns over several vibrational quantum numbers for this observable are also predicted for each excited state [8] which, to-date, cannot be experimentally verified.
It is a pleasure for the author of being invited to contribute to this book as a tribute to Gaston Berthier who taught him in the late sixties at 'Ecole Normale Supérieure (rue Lhomond, Paris)' how to use a particular molecular orbital formalism, developped in his group, for a study on transiton metal complexes. This has been the beginning of a fruitful collaboration over the years.
This work was performed as part of the research programme of the 'Foundation for Fundamental Research on Matter (FOM)' which is financially supported by the 'Netherlands Organization for the Advancement of Scientific Research (NWO)'.
References
1.V.J. Curry, L and G. Herzberg, Z.Phys. 86, 348 (1933).
2.F.C. Wyze, E.L. Manson and W. Gordy, J.Chem.Phys. 57, 1106 (1972).
3.L.M. Ziuris, Astrophys.J. 321, L81 (1987).
4.S.N. Ghosh, R.D. Verma and J. VanderLinde,Canad.J.Phys. 59, 1640 (1981). R.A. Gottscho, R.W. Field and H. Lefebvre-Brion, J.Mol.Spectr. 70, 420(1978).
5.B. Coquart and J.C. Prudhomme, J.Mol.Spectrosc. 87, 75 (1981).
6.R.D. Verma, S.N. Ghosh and Z. Iqbal, J.Phys.B. 20, 3961 (1987).
7.G. de Brouckère, D. Feller, J.J.A. Koot and G. Berthier, J.Phys.B. 25, 4433
(1992).
8.G. de Brouckère, D. Feller and J.J.A. Koot , J.Phys.B, in press.
9.L. McMurchie, S. Elbert, S. Langhoff and E.R. Davidson "MELDF Suite of Programs" substantially modified by D. Feller, R.J. Cave, D. Rawlings, R.
Frey, R. Daasch,
L. Nitzche, P. Phillips, K. Iberle, C. Jackels and E.R. Davidson (1990/1991).
10.D. Feller and E.R. Davidson, J.Chem.Phys. 74, 3977 (1981).
11. E.R. Davidson and D.W. Silver, Chem.Phys.Lett. 52, 403 (1977).
12.S.R. Langhoff, C.W. Bauschlicher and P.R. Taylor, Chem.Phys.Lett. 180, 88 (1991).
13.J.L. Dunham, Phys.Rev. 41, 721 (1932).
14.The suite of programs VIBRQT (Vibration-rotation analysis program for diatomic
molecules) was written by T.H. Dunning, Molecular Science Research Center, Pacific Northwest Laboratory, Richland, 1978/1979, and modified by R. Eades (1983) and D. Feller (1990).
15.F. Grein and A. Kapur,J.Mol.Spectrosc. 99, 25 (1983).
16.G.H. Fuller, J.Phys.Chem.Ref.Data 5,835(1976).
17.A.D. McLean and M. Yoshimine, IBM J.Res.Dev. 12, 206 (1968).
18.J. Raymonda and W. Klemperer,J.Chem.Phys. 55, 232 (1971).
19.M. Larsson, Chem.Phys.Lett. 117, 331 (1985).
20.The suite of programs INTENSITY was written by W.T. Zemke and W.C. Stwalley, Department of Chemistry, Wartburg College, (1978)
21. The program SPLINE was written by K.A. Kaiser, Data processing, Southern Illinois University, (1978).
332 |
G. DE BROUCKERE |
22.K.P. Huber and G. Herzberg, Molecular Spectra and Molecular Structure. IV. Constants of Diatomic Molecules,Van Nostrand Reinhold, New York (1979). G. Herzberg, Molecular Spectra and Molecular Structure. I. Spectra of Diatomic Molecules (p. 154), Van Nostrand, New York (1961).
23.M.B. Moeller, M.R. McKeever and S.J. Silvers, Chem. Phys.Lett. 31, 398 (1975)
24.M.E. Rozenkrantz and K.Kirby, J.Chem.Phys. 90, 6528 (1989).
25.M.B. Moeller and S.J. Silvers, Chem. Phys.Lett. 19, 78 (1973).
Theoretical Treatment of State-Selective Charge Transfer Processes.
He as a Case Study
M.C. BACCHUS-MONTABONEL
Laboratoire de Spectrométrie Ionique et Moléculaire (URA CNRS n°171),
Université Lyon I, 43 Bd du 11 Novembre 1918, 69622 Villeurbanne, France
1. Introduction
Charge transfer recombination of multiply charged ionic species in collision with neutral atoms or molecules is an important process in astrophysical plasmas [1,2] and controlled nuclear fusion research [3]. From an experimental point of view, these reactions have been extensively studied in recent years using a wide variety of techniques (VUV spectroscopy [4-11], energy gain spectroscopy [12,13], electron spectrometry [14,15]).
Much attention has also been paid to the interpretation of the electron capture processes using model potential methods [16-181 which allow generally a fair description of the phenomena in the case of closed shell systems, or ab initio methods [19-24] necessary for the study of open-shell systems as for example low-charged ions or metastable states.
Recently, we have developed a full theoretical treatment of electron capture processes involving an ab initio molecular calculation of the potential energy curves and of the radial and rotational couplings followed, according to the collision energy range concerned, by a semi-classical [21-23] or quantal [24] collision treatment.
As a test case, we report in this paper the study of the He collision. This work has been undertaken in connection with photon spectroscopy experiments regarding the electron capture for the reactions
at collision energies in the range [10-100 keV] [4,7].
In accordance with previous investigations [8,9], these experiments have shown a quite
different behaviour for |
than for other multicharged ions such as the isoelectronic |
|
ion |
. The single-electron capture process has been shown to be dominant on the |
|
n = 3 levels and in particular on the 3s level for collision energies lower than 50 keV. A high probability of double capture has also been observed characterized by an intense peak at nm attributed to the transition [4,5,7]. Furthermore, for this system, metastable levels can be populated by foil excitation and may thus be present in the incident beam. Experimental measurements have been performed for the
333
Y. Ellinger and M. Defranceschi (eds.), Strategies and Applications in Quantum Chemistry, 333–348.
© 1996 Kluwer Academic Publishers. Printed in the Netherlands.
334 |
|
|
and |
He collisions at 60 keV from photon spectroscopy |
|
[6] and at 51 keV from electron spectroscopy [14]. |
||
A complete theoretical |
treatment of the |
He collision should therefore take into |
account, first the single-electron capture process from the ground state entry channel
and also from the metastable level |
in order |
|
to take care of the fraction of metastable |
ion in the incident beam, as well as the |
|
double-electron capture process from both ground and metastable |
ions. |
|
The single-electron capture process from the ground state |
is the |
|
easiest one to handle and also the most important one. The capture being dominant on the n = 3 levels, and the effect of spin-orbit coupling being of negligible importance for electron capture in the energy range of interest, we have determined the potential energy
curves corresponding to the entry channel |
and all the |
|
and |
states corresponding to the |
configuration. |
The consideration of the collision involving the metastable ion
requires the calculation of much higher levels. The work has been undertaken in tight connection with experimental investigations in order to reduce the number of states involved in the molecular calculation. From an experimental point of view, it is assumed that only the triplet metastable state will be involved in the collision because of the shorter lifetime of the singlet state [25] with respect to the time-of-flight of the ions from their production to the collision cell. Besides, among the doublet and quartet states
produced in the single-electron capture process, the doublet states are rapidly autoionized, when the quartet states are metastable with respect to Coulomb autoionization and then only transitions involving these quartet states may be observed.
As in the collision of the ground state ion on a He target, the main process has been shown experimentally to be the core-conserving single-electron capture on the n = 3 levels [6,14] with a small amount of capture on the n = 4 levels. The transfer-excitation process corresponding to a single-electron capture and an excitation of the core leading to
states has also been observed, with a dominance of the capture on the levels [6,10]. In view of all these experimental findings we have thus considered the collisional processes
and calculated the potential energy curves for the |
entry channel and the |
and |
||
states of the |
configuration as well as the |
and |
states |
|
corresponding to the |
configuration which could partly account |
|||
for the transfer-excitation process. |
|
|
|
|
For the double-electron capture |
|
|
a very large |
|
energy gap separates the two potential energy curves involved in the process, thus many molecular curve crossings may be important and the population of the
level should probably come from a cascade effect. In practice, a complete treatment including all the potential energy curves is impossible [18,26].
THEORETICAL TREATMENT OF STATE-SELECTIVE CHARGE-TRANSFER PROCESSES |
335 |
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As this process has been shown experimentally [5] to be dominant on the |
|
|||
level, we have performed a calculation including the entry channel |
|
|||
and the |
and |
states corresponding to the |
|
configu- |
ration. We have also considered the |
states which are energetically |
|||
very close to the double capture states. Such a calculation could certainly hardly provide quantitative results, but it could give some qualitative information on the behaviour of the collisional system.
It has otherwise been shown experimentally [5] that the double-electron capture occurs
mainly from the |
, |
. |
ground state. We have thus neglected the contribu- |
tion of the |
metastable ion. |
|
|
The electron capture processes are driven by non-adiabatic couplings between molecular states. All the non-zero radial and rotational coupling matrix elements have therefore been evaluated from ab initio wavefunctions.
2. Computational method
The potential energy curves have been determined by ab initio calculations with configuration interaction according to the CIPSI algorithm [27]. The SCF calculation has been performed by means of the Psondo program [28,29] for the electronic configuration
From a molecular point of view, the and the He collisional systems have to be considered separately.
For the ground state system compact configuration interaction (CI) spaces have been used in the calculation (about 100-150 determinants) with a threshold
for the contribution to the perturbation. According to the deep energy difference between and the molecular orbital has been frozen in the CI procedure. The basis of atomic functions used in the calculation [21] is a 9s5p3d basis of gaussian functions for nitrogen and a 4slp basis for helium optimized from the 6-311G** basis of Krishnan et al. [30J. Diffuse functions have been added — 2s2pld for nitrogen and 1p for helium — and optimized with respect to the experimental data by means of a one-configuration
calculation for the excited states of and respectively.
This basis leads to a reasonable agreement with experiment [31] for a large number of atomic levels of nitrogen (Table 1). For the determination of the couplings between the states involved in the double electron capture process, a less expanded basis set has been chosen — 8s4p3d basis set for nitrogen and 3s1p basis set for helium — leading to shorter computation time while saving a fairly reasonable agreement with experiment.
The calculation performed for the metastable He system has necessitated somewhat larger CI spaces (200-250 determinants) in order to reach the same perturbation threshold the molecular orbital being not frozen for this calculation.The basis of atomic orbitals has been also expanded to a 10s6p3d basis of gaussian functions for
nitrogen reoptimized on |
for |
the s exponents and on |
for the p |
exponents and added of one s and one p |
diffuse functions [22]. For such excited states, |
||